A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces

When the set of closed subspaces of C(Delta), where Delta is the Cantor set, is equipped with the standard Effros-Borel structure, the graph of the basic relations between Banach spaces (isomorphism, being isomorphic to a subspace, quotient, direct sum...) is analytic non-Borel. Many natural families of Banach spaces (such as reflexive spaces, spaces not containing l(1) (w),...) are coanalytic non-Borel. Some natural ranks (rank of embedding, Szlenk indices) are shown to be coanalytic ranks. Applications are given to universality questions. Analogous results are shown for basic sequences modulo equivalence.